On the Bennett-Hoeffding inequality
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, p. 15-27

The well-known Bennett-Hoeffding bound for sums of independent random variables is refined, by taking into account positive-part third moments, and at that significantly improved by using, instead of the class of all increasing exponential functions, a much larger class of generalized moment functions. The resulting bounds have certain optimality properties. The results can be extended in a standard manner to (the maximal functions of) (super)martingales. The proof of the main result relies on an apparently new method that may be referred to as infinitesimal spin-off. Parts of the proof also use the method of certificates of positivity in real algebraic geometry.

La borne de Bennett-Hoeffding pour des sommes de variables aléatoires indépendantes est précisée, en prenant en compte la partie positive des troisièmes moments et sensiblement améliorée en utilisant, au lieu de la classe de toutes les fonctions exponentielles croissantes, une classe beaucoup plus important de fonctions de moment généralisées. Les limites qui en résultent ont certaines propriétés d'optimalité. Les résultats peuvent être étendus de manière standard pour (les fonctions maximales de) (sur)martingales. La preuve du résultat principal repose sur une méthode apparemment nouvelle. Des éléments de la preuve utilisent également la méthode des certificats de positivité de la géométrie algébrique réelle.

DOI : https://doi.org/10.1214/12-AIHP495
Classification:  60E15,  60G50,  60E07,  60E10,  60G42,  60G48,  60G51
Keywords: probability inequalities, sums of independent random variables, martingales, supermartingales, upper bounds, generalized moments, Lévy processes, certificates of positivity, real algebraic geometry
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     author = {Pinelis, Iosif},
     title = {On the Bennett-Hoeffding inequality},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {1},
     year = {2014},
     pages = {15-27},
     doi = {10.1214/12-AIHP495},
     zbl = {1288.60025},
     mrnumber = {3161520},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_1_15_0}
}
Pinelis, Iosif. On the Bennett-Hoeffding inequality. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 15-27. doi : 10.1214/12-AIHP495. http://www.numdam.org/item/AIHPB_2014__50_1_15_0/

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